Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics

2010/02/20 by Vladimir S. Matveev
Mathematics · Physics and Astronomy · #math.DG #math-ph #math.MP

paper · pdf · doi:10.2969/jmsj/06410107

published as J. Math. Soc. Jpn. 64(2012), no. 1, 107--152 · 33 pages, 7 pictures. This paper replaces arXiv:0911.3521v1 and arXiv:1002.0133

arxiv created 2010/02/20 · arxiv updated 2013/01/22

Abstract

We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture.

Citations

• Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta
• Proof of projective Obata conjecture for two-dimensional pseudo-Riemannian metrics